Finite element analysis (FEA) is widely used to model and solve engineering problems relating to complex systems such as three-dimensional non-linear structural design and analysis. FEA derives its name from the manner in which the geometry of the object under consideration is specified. Basically, FEA computer software is provided with a model of the geometric description and the associated material properties at each point within the model. In this model, the geometry of the system under analysis is represented by solids, shells and beams of various sizes, which are called elements. The vertices of the elements are referred to as nodes. The model is comprised of a finite number of elements, which are assigned a material name to associate the elements with the material properties. The model thus represents the physical space occupied by the object under analysis along with its immediate surroundings. The FEA software then refers to a table in which the properties (e.g. stress-strain constitutive equation, Young's modulus, Poisson's ratio, thermo-conductivity) of each material type are tabulated. Additionally, the conditions at the boundary of the object (i.e., loadings, physical constraints, etc.) are specified. In this fashion a model of the object and its environment is created.
A requirement for successful FEA is a finite element mesh which reproduces the component accurately and in particular has sufficient density of elements at critical locations to model essential features of the problem, but not so high a density of elements that unnecessary detail is modelled and the computer processor is overburdened. Further, the elements should exhibit acceptable quality measures such as low skewnesses, low aspect ratios, and smooth size transitions from element to element. Particularly in respect of complex-shaped components, such as gas turbine engine (GTE) blades and vanes, a considerable amount of effort can thus go into meshing the component.
A typical process for developing and manufacturing a component involves iterative rounds of computer aided design (CAD) and FEA, until a final design is arrived at that can go into testing or manufacture. However, due to manufacturing variability and tolerances the actual manufactured examples may differ in shape from the nominal shape of the FEA mesh used during the development phase. The mesh can be considered as a generic representation of a nominal component, with actual manufactured examples of the component displaying shape variation relative to that generic mesh. The shape variation can be measured by various known metrology techniques such as optical (e.g. laser) scanning, touch probe measurements, computed tomography etc.
If the variations are small, then the differences between the generic mesh and the actual components may be immaterial. However, if the variations are not small a problem can then arise that FEA performed on the generic mesh may not accurately model the actual components. A similar problem occurs if it is wanted to use FEA to quantify the effect of shape variations between actual components. For example, if a frequency prediction of a property of a manufactured component differs from a test of that property, it is helpful to know how much of that difference can be attributed to nominal versus manufactured shape deviations.
One approach to address the problem is simply to distort the generic mesh to the shape of the actual component. However, a simple distortion can reduce the quality of the mesh, e.g. by increasing element skewnesses and aspect ratios, and disturbing smooth size transitions. Another approach is not to use the generic mesh to model the actual component, but rather to re-mesh each time an actual component is to be modelled. However, re-meshing can be labour intensive and time-consuming. Further, for valid comparisons between analyses of different actual components it is desirable that the meshes for the different analyses are fundamentally similar rather than each being the result of a fresh re-mesh.